High-accuracy Runge–Kutta methods for stiff ODE systems

Ordinary differential equations (ODEs) that are stiff come up a lot in science and engineering. They need numerical methods that balance stability, speed, and accuracy. In strict regimes, explicit Runge–Kutta schemes don’t always work, and implicit schemes are very hard to compute. This study creates high-precision Runge–Kutta methods that focus on L-stability and order conditions to make sure they are stable. To get the best results when handling stiff problems, modified stability-preserving formulas and hybrid implicit-explicit (IMEX) methods are suggested. These methods give us a solid way to solve complicated stiff ODE systems that happen in the real world and cover many areas.